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Theorem prmoval 15661
Description: Value of the primorial function for nonnegative integers. (Contributed by AV, 28-Aug-2020.)
Assertion
Ref Expression
prmoval (𝑁 ∈ ℕ0 → (#p𝑁) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1))
Distinct variable group:   𝑘,𝑁

Proof of Theorem prmoval
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 df-prmo 15660 . . 3 #p = (𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
21a1i 11 . 2 (𝑁 ∈ ℕ0 → #p = (𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1)))
3 oveq2 6612 . . . 4 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
43prodeq1d 14576 . . 3 (𝑛 = 𝑁 → ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1))
54adantl 482 . 2 ((𝑁 ∈ ℕ0𝑛 = 𝑁) → ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1))
6 id 22 . 2 (𝑁 ∈ ℕ0𝑁 ∈ ℕ0)
7 prodex 14562 . . 3 𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1) ∈ V
87a1i 11 . 2 (𝑁 ∈ ℕ0 → ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1) ∈ V)
92, 5, 6, 8fvmptd 6245 1 (𝑁 ∈ ℕ0 → (#p𝑁) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3186  ifcif 4058  cmpt 4673  cfv 5847  (class class class)co 6604  1c1 9881  0cn0 11236  ...cfz 12268  cprod 14560  cprime 15309  #pcprmo 15659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-iota 5810  df-fun 5849  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-seq 12742  df-prod 14561  df-prmo 15660
This theorem is referenced by:  prmocl  15662  prmo0  15664  prmo1  15665  prmop1  15666  prmdvdsprmo  15670  prmolefac  15674  prmodvdslcmf  15675  prmgapprmo  15690
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